Flow of non-Newtonian fluids through fixed beds of particles: Comparison of two models

Chemical Engineering and Processing 37 (1998) 169 – 176

Flow of non-Newtonian fluids through fixed beds of particles: Comparison of two models I. Machac' a,*, J. Cakl a, J. Comiti b, N.E. Sabiri b a b

Department of Chemical Engineering, Uni6ersity of Pardubice, 53210 Pardubice, Czech Republic Laboratoire de Ge´nie des Proce´de´s, I.U.T. Saint-Nazaire, BP 420, 44606 Saint-Nazaire, France Received 11 April 1997; received in revised form 17 August 1997; accepted 29 August 1997

Abstract Two models based on modifications of the capillary representation of a particle bed are proposed in order to evaluate the pressure gradient for the flow of non-Newtonian purely viscous fluids through fixed beds. In the first model, the ideas of flow around particles are considered simultaneously and the bed factor is used as a dynamic characteristic of the bed. In the second one, the tortuosity factor and dynamic specific area are used as bed structure parameters. The validity and accuracy of the models have been tested using an extensive set of experimental data concerning fixed beds of spherical and various shaped non spherical particles. In these experiments, 19 particle-fluid systems were investigated in the range of Reynolds number including creeping and transition flow regimes; power-law fluid behaviour index was ranged between 0.27 and 1. The mean deviation between pressure drop model predictions and the whole set of experimental data is less than 10% for both models presented. The models are efficient even when important wall effect exists. © 1998 Elsevier Science S.A. All rights reserved. Keywords: Non-Newtonian fluid flow; Power-law; Pressure drop; Fixed beds; Spherical and non spherical particles; Creeping and inertial flow regime

1. Introduction Flow of a non-Newtonian fluid through a fixed bed of particles is applied in a number of processes of chemical and related industries (e.g. filtration of polymeric liquids and slurries, heterogeneous catalysis using fixed bed reactors, oil recovery by polymer flooding, and others). A lot of experimental and theoretical studies have been concerned with pressure drop determination for non-Newtonian fluid flow through porous media; most of them have been reviewed by Kemblowski [1] and Chhabra [2]. For the sake of simplicity, the equations for non-Newtonian fluid pressure drop prediction are most frequently derived on the basis of a simplified capillary bed model in which a granular bed structure is modelled as a bundle of parallel capillary tubes, and most of them concern only fixed beds of spherical particles [3 – 7]. The specificity of the structure of various fixed beds of particles leads to the effort to * Corresponding author. [email protected]

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propose a more general bed model in order to relate pressure drop to parameters characterising the bed structure. In this contribution, two simple methods are presented for the pressure drop prediction of the flow of purely viscous non-Newtonian fluids through fixed beds of spherical and non-spherical particles. The suggested predictive relationships are based on two different modifications of the capillary bed model. The validity and accuracy of these relationships have been tested experimentally for the flow of power-law fluids through fixed beds of spherical and non-spherical particles in creeping and transition flow regimes. 2. Pressure-drop relationships In order to provide a single presentation of the two models, the approach of Machac' and Dolejs' [8] is used. The application of a capillary bed model is based on the fact that the bed is considered as a bundle of model capillaries and relationships describing the fluid flow through a bed are expected to be analogous with those valid for the fluid flow through a model capillary.

I. Machac' et al. / Chemical Engineering and Processing 37 (1998) 169–176

170

A general form of the equation for a laminar flow of a purely viscous non-Newtonian fluid through a capillary of circular cross-section may be written as Dw =

&

4 t 3w

tw

t 2g; (t) dt

dynamic variable tpw, which is supposed to represent the mean shear stress at the solid surface, and the kinematic variable Dpw, which is related to corresponding mean shear rate g; pw, can be expressed as

(1) tpw1 =

0

where Dw =

(8)

2u lcho

(9)

and

2uch 4V: = 3 lch pR

(2) Dpw1 =

is the kinematic variable which is related to wall shear rate g; w, and tw =

DP l L ch

DP DPR l = Lch ch 2Lc

(3)

The principal difference between a classical capillary bed model and this approach is in the determination of the linear dimension lch of the void space in the bed. Here, it is expressed on the basis of the requirement that relationship (8) must fulfil the momentum balance for the flow through the bed. It follows from the simple scalar form of this balance that

is the dynamic variable identical with the wall shear stress, uch is the characteristic fluid velocity, lch is the characteristic linear dimension of the capillary cross section, Lch is the characteristic length of the capillary, t is the shear stress, g; is the shear rate, DP is the pressure drop, V: is the volume flow rate, R is the radius of the capillary, and Lc is the length of the capillary. A particular form of relationship (Eq. (1)) can be obtained by its integration using a relevant flow model of the fluid. Thus, for the power law model

where aps is the specific surface of particles (static specific surface area), aw is the specific surface of the apparatus walls, 8 is the bed factor, which is defined as a ratio of total and friction drags of the bed, while

g; =(t/K)1/n

dp = 6/aps

(4)





with parameters n and K, we have tw = K

3n+1 Dw 4n

n

(5)

Relationship (5) is expected to be also valid for a laminar flow of a power law fluid through a fixed bed of particles if the variables tw and Dw are substituted by corresponding bed variables tpw and Dpw. Two different concepts used in our laboratories for derivation of predictive fixed bed pressure drop equations [8–10] are described below. The difference between these concepts lies in different definition of bed characteristic quantities uch, lch, and Lch, and in a different way of taking into account the inertial resistance of the bed.

2.1. Equation based on the concept of a bed factor (model I) In the approach of Machac' and Dolejs' [8], the quantities uch and Lch were defined for a fixed bed flow as uch =uch1 =

u o

(6)

lch = lch1 =

(7)

Here u is the superficial velocity of the liquid, o is the bed voidage, L is the length of the bed. Then, the

(10)

(11)

is the effective diameter of particles, and M1 the corrective coefficient for the wall effect. The bed factor 8 corresponds to the ratio of the total to the friction drag on the particles being in the bed. It was found that the bed factor 8 is independent on the rheological behaviour of purely viscous fluids. However, it depends on the shape and orientation of particles in the bed, and in the transition flow regime on a Reynolds number as well. The most reliable way to obtain the bed factor of a given bed is the experimental one using a Newtonian fluid [8,10]. The coefficient M1 is given by the expression M1 = 1+

2dp 38(1− o)Dh

(12)

where Dh = 4/aw is hydraulic diameter of the apparatus. The linear dimension lch1 may also be considered as a modified bed hydraulic radius in which the wetted surface of the bed is enlarged by an additional surface of particles, whose friction drag is the same as the form drag of the actually wetted surface. The pressure drop equations are often presented in a dimensionless form, for example as a function of the bed friction factor

and Lch =Lch1 =L

o dpo = aps8(1− o)+ aw 68(1−o)M1

fp =

DP dp o 3 ru 2 L (1−o)

(13)

on a Reynolds number. If the Reynolds number defined for a power law fluid flow by the expression

I. Machac' et al. / Chemical Engineering and Processing 37 (1998) 169–176

Repn =



ru 2 − n d np o 2(n − 1)121 − n 4n n K (1 −o) 3n +1



n

(14)

is used, Eq. (5) with variables tpw1 and Dpw1 can be rewritten into the form fp =

72(8M1)n + 1 Repn

(15)

Eq. (15) is expected to be valid for the flow of a power law fluid through fixed beds of spherical and non-spherical particles in both creeping and transition flow regimes.

2.2. Equation based on the concept of the tortuosity factor and the dynamic specific surface area (model II) The authors Sabiri and Comiti [11] expressed the bed characteristic quantities Lch and uch as Lch =Lch2 =tL uch =uch2 =

ut o

(16) (17)

Here t is the tortuosity of the cylindrical model pore. The characteristic length lch is considered to be equal to the hydraulic radius of the pores in the bed, so that lch = lch2 =

o apd(1−o)+ aw

(18)

where apd is the actually wetted specific surface of particles (dynamic specific surface area). The bed resistance over a range of Reynolds number including creeping and inertial flow regimes is considered to be the sum of viscous resistance and inertial resistance terms. The viscous part of the resistance can be evaluated according to Eq. (5) with dynamic and kinematic variables defined as tpw2 =

DP lch2 Lt

(19)

2ut lch2o

(20)

and Dpw2 =

Then, the friction factor of the bed, designated for the creeping flow regime as fpc is given by fpc =

72(tXM2)n + 1 Repn

(21)

apd aps

(22)

is the fraction of the particle surface area in the bed offered to the flow, and

2dp 3X(1− o)Dh

(23)

is a corrective coefficient of the wall effect for the viscous term. The second (inertial) term of the resistance to flow is independent on the fluid viscosity, and then it is assumed to be the same as that proposed for Newtonian fluids by authors Comiti and Renaud [12]. Hence, the general equation for the friction factor fp may be written as fp =

72(tXM2)n + 1 + 0.58t 3XN2 Repn

where

 

N2 = 1− 1−

n

dp Dh

2



0.427+ 1−

(24)

dp Dh



2

(25)

is the corrective coefficient of the wall effect for the inertial term. Like the bed factor 8, the structural parameters t and X for fixed beds are independent of the rheological behaviour of the fluid, and can be obtained from pressure drop measurements with a Newtonian fluid [12,11].

3. Mean shear rate estimation The shear stress–shear rate dependence (flow curve) of a purely viscous non-Newtonian fluid can be described with a sufficient accuracy by a power law (Eq. (4)) only in a limited interval of the shear rate g; . For a flow through a fixed bed of particles, this drawback can be eliminated if the total range of shear rate investigated is divided into two or more intervals and the power law parameters K and n are determined from rheological data in such an interval of the shear rate which corresponds to that reached in the fixed bed flow. It follows from the comparison of the power law model (Eq. (4)) with Eq. (7) that the mean value of the shear rate g; pw at the solid surface in the bed can be expressed as g; pw =

3n+ 1 Dw 4n

(26)

If the wall effect is neglected (aps  aw), we have g; pw1 =

Here X=

M2 = 1+

171

3n+ 1 (1−o) aps8 2n o2

(27)

3n+ 1 (1−o) apstX 2n o2

(28)

or g; pw2 =

I. Machac' et al. / Chemical Engineering and Processing 37 (1998) 169–176

172

Table 1 Power law parameters of non-Newtonian fluids used Fluid

r (kg m−3) n

K (Pa sn)

CMC 0.6%

1022

999 999

0.912 0.771 0.634 0.741 0.269

0.0644 0.116 0.262 0.068 0.865

2.5–60 38–450 400–3500 100–8000 30–80

999

0.407 0.625 0.428

0.462 0.117 0.242

100–800 900–10000 30–85

0.534 0.711

0.129 0.039

100–950 1000–10000

Natrolsol 0.5% Xanthan 0.22%

Xanthan 0.14%

The viscosity function of these polymer solutions was represented by a series of power law parameters n and K evaluated from the flow curve data in the range of shear rates corresponding to those reached in the fixed bed experiments (Table 1). The fixed beds packed in a cylindrical column were composed of spheres, cylinders, polyhedrons (randomly packed), and square based plates (tightly packed). The geometrical characteristics of particles (specific surface aps, diameter d, height h, side length a, and width e) are given in Table 2 along with bed characteristics o, 8, t, and X. The experimental conditions are presented in Table 3. The experiments are described in more details elsewhere [13,14].

Range of g; (s−1)

5. Results and discussion 4. Experimental

5.1. Bed characteristics For testing the proposed relationships, two sets of pressure drop data have been exploited, which were measured at Pardubice laboratory (set 1) and at SaintNazaire laboratory (set 2). In the experiments, the pressure drop during the flow of water and solutions of glycerol (Newtonian fluids), and during the flow of water solutions of polymers (non-Newtonian fluids) through fixed beds of spherical and non-spherical particles was measured as a function of the liquid flow rate. Shear thinning water solutions of hydroxyethylcellulose (1.5 wt.% Natrosol 250 MR) and polysaccharide (0.14 and 0.22 wt.% Xanthane CX 12) were used in the first set of experiments, and a shear thinning water solution of carboxymethylcellulose sodium salt (0.6 wt.% BDH high viscosity) was used in the second one.

Making use of pressure drop measurements with Newtonian fluids (solutions of glycerol and water), the values of the bed factor 8 and the parameters t and X were evaluated for individual fixed beds from Eq. (15) or from Eq. (24). Experimental dependences of the bed factor on the Reynolds number, which are expected to be the same for any purely viscous fluid flowing through the given bed, can simply be described by a second order polynomial 8=80 + 81Rem + 82Re2m

(29)

Here, the Reynolds number Rem as a measure of the ratio of inertial to the viscous forces acting on the fluid, is expressed for a power law fluid as

Table 2 Geometrical characteristics of particles and structural parameters of fixed beds investigated Type of particles

Spheres Spheres Spheres Spheres Polyhedrons Cylinders Cylinders Plates Plates

Geometrical characteristics (10−3 m)

d=2.92 d=1.46 d=1.78 d=1.92 d=4.0 h=2.7 h=3.3 h/d= 1.22 h=5.49 h/d= 5.49 e= 0.52 e/a= 0.102 e= 1.05 e/a= 0.209

aps (m−1)

8

t

X = apd/aps o

80

81

82

2055 4096 3371 3120 2074

1.453 1.453 1.453 1.453 1.655

0.058 0.058 0.058 0.058 0.175

0.012 0.012 0.012 0.012 0.019

1.44 1.41 1.41 1.37 1.89

1 1 1 1 0.69

0.36 0.38 0.38 0.37 0.38

1966

1.384

0.166

0.000

1.66

0.81

0.37

4368 4368 4659 4659 2713

1.523 1.523 1.781 1.781 1.438

0.290 0.290 1.043 1.043 0.731

0.011 0.011 0.211 0.211 0.02

1.9 1.9 3.49 3.49 2.77

0.80 0.80 0.5 0.5 0.53

0.39 0.39 0.46 0.46 0.35

I. Machac' et al. / Chemical Engineering and Processing 37 (1998) 169–176

Rem =





173

ru 2ch ru 2 − nd np o 2(n − 1) 4n n = (8M1) − n n n tpw K (1 − o) 12 3n +1

=Repn

1 12(8M1)n

(30)

The values of the parameters 80, 81, and 82 obtained by nonlinear regression of experimental dependences 8 = 8(Rem) are summarised in Table 2 along with parameters t and X for the individual fixed beds tested. The polynomial parameter 80 represents the value of bed factor in the creeping flow regime. If the value of Rem, at which the value of the bed factor 8 exceeds the value of 80 by 5%, is considered as a limiting value of Rem for the creeping flow regime, than it follows from Eq. (29) that the limiting value of Rem depends on the particle shape (from 0.1 for tightly packed plates to 1 for random packed spheres).

5.2. Comparison of experimental and calculated pressure drop data To evaluate the accuracy of the proposed methods for the pressure drop prediction, the mean relative deviations d=

1 m % di m i=1

(31)

of experimental pressure drop data and those calculated from Eqs. (15) and (24) were determined for each system investigated. Here m is the number of experimental data points and

)

di = 1−

(DPcal)i (DPexp)i

)

(32)

The values of d obtained for the individual systems and both of the predictive methods are given along with the range of deviations di in Table 3. One can see that the agreement between the model predictions and the experimental values is satisfactory especially for the beds packed with non spherical particles. In this case, the relative error does not exceed the value of 15%. The prediction based on Eq. (15) leads to a slightly greater scatter between calculated and experimental pressure drop data as compared with the prediction based on Eq. (24). On the other hand, the mean deviation d calculated from all experimental data is 8.8% for Eq. (15) and 9.4% for Eq. (24). The good agreement between model predictions and experimental data is clearly shown in Fig. 1(a) and (b), where the experimental values of the friction factor fp are compared with those calculated using Eqs. (15) and (24). The experimental pressure drop data have also been compared with those calculated according to relationships given in the literature. The first set of experiments has been compared (Machac' and Cakl [9]) with the data calculated using equations proposed by Kemblow-

Fig. 1. Comparison of experimental bed friction factor data with calculated ones: (a) Eq. (15); (b) Eq. (24). , cylinders; h/d=1.22;

, cylinders, h/d = 5.94; , plates, e/a =0.209; , plates, e/a = 0.102; “, spheres, 2.92 mm; , spheres, 1.92 mm; , spheres, 1.78 mm; , spheres, 1.46 mm; , polyhedrons.

ski and Mertl [3], Mishra et al.[5], Brea et al. [6], Michele [7], and Kumar et al. [15], the second set of experimental pressure drop data [11] has been compared with those calculated according to equations proposed by Kumar et al. [15], and Chhabra and Srinivas [16]. It was found that the all equations tested lead, especially for the flow through the beds of nonspherical particles in the transition flow regime, to greater deviations d in comparison with those given in Table 3. Examples of the comparison between the experimental pressure gradients and their predictions calculated as a function of superficial velocity according to Eqs. (15) and (24), and according to equations proposed by Kumar et al. [15], and Chhabra and Srinivas [16] are shown for some systems tested in Fig. 2(a)–(d). This comparison shows that the equation of Kumar et al. leads to the worst results. The equation of Chhabra and

Particles

Plates e/a=0.209 Plates e/a= 0.102 Cylinders h/d= 5.49 Spheres 2.92 mm

Spheres 1.46 mm Spheres 1.78 mm Spheres 1.92 mm Polyhedrons h/d=0.67

Spheres 1.46 mm Spheres 1.78 mm Spheres 1.92 mm Polyhedrons h/d=1.22 Cylinders h/d= 1.22

Spheres 1.46 mm Spheres 1.78 mm Spheres 1.92 mm Polyhedrons h/d= 0.67 Cylinders h/d=1.22

Fluid

CMC 0.6%

Natrolsol 0.5%

Xanthan 0.22%

Xanthan 0.14%

0.111–148 0.061–208 0.09–208 0.105–342 0.399–320

0.032–80.6 0.007–124 0.022–135 0.062–215 0.027–236

0.160–24.7 0.316–43.9 1.64–43.9 0.339–79.1

0.109–7.46 0.057–8.30 0.103–7.76 0.169–27.2

Range of Repn

98 – 18420 54 – 21397 61 – 19200 45 – 24360 97 – 17017

76 – 13550 30 – 16405 54 – 16215 68 – 19719 38 – 17841

98 – 5708 126 – 7180 434 – 6663 70 – 7560

73 – 4837 60 – 3960 120 – 4852 66 – 4082

5.1 6.1 4.7 9.9 5.7

5.9 4.7 6.0 11.2 6.5

18.2 18.1 10.0 8.7

8.8 4.8 3.1 15.9

0.2 – 10.5 1.9 – 13.5 0.2 – 15.8 3.1 – 24.5 0.5 – 16.7

0.4 – 13.7 0.3 – 10.0 0.3 – 11.2 4.0 – 21.7 2.0 – 15.3

0.6 – 27.6 7.6 – 23.5 4.3 – 19.2 1.4 – 16.5

2.4 – 13.2 1.0 – 11.7 0.6 – 8.1 9.8 – 25.4

94 – 11187 52 – 11187 59– 9987 34– 3402 81– 6443

86– 9940 29 – 9792 51 – 9887 52– 6667 32– 7304

94– 5123 120– 6032 411 – 5581 53 – 4049

72– 3500 57– 3100 108 – 3478 60– 3500

Range of g; (s−1)

Range of relative errors di (%)

Range of g; (s−1) Mean relative error d (%)

Model II

Model I

Table 3 Experimental conditions and values of the scatter between the model predictions and the experimental results

10.3 10.3 6.5 5.2 10.5

6.6 12.6 10.2 9.5 13.3

13.7 13.9 8.8 13.9

3 5.7 9.3 10

0.1–21.1 0.1–21.1 0.4–10 0.5–19.3 0.9–17.9

0.3–11.8 0.9–21.9 0.75–17.8 0.9–17.6 2.3–22.2

0.1–18.5 5.6–16.8 6.6–11.6 5.4–19.2

0.2–9 0.3–14 2–14 0.4–19

Mean relative error d Range of relative errors (%) di (%)

174 I. Machac' et al. / Chemical Engineering and Processing 37 (1998) 169–176

I. Machac' et al. / Chemical Engineering and Processing 37 (1998) 169–176

175

Fig. 2. Comparison of experimental pressure gradients with predicted ones as a function of superficial velocity: (a) flow of CMC (0.6 wt.%) through bed of plates (e/a= 0.102); (b) flow of Natrosol (0.5 wt.%) through bed of polyhedrons; (c) flow of Xanthane (0.14 wt.%) through bed of spheres (d =1.92 mm); (d) flow of Xanthane (0.22 wt.%) through bed of cylinders (h/d =1.22). , experimental data; — , Comiti et al. (Eq. (24)); ——, Machac' et al. (Eq. (15)); ——, Kumar et al; - - - Chhabra et al.

Srinivas gives satisfactory results in the creeping flow and in the transition regime only for beds of particles, for which the form drag is not too different from that of spherical particle beds. It can be seen in Fig. 2(a) that the evaluation of inertial contribution is inadequate for beds of tightly packed plates.

Especially for non-spherical particles and transition regime, the proposed models lead to more satisfactory pressure drop predictions in comparison with those calculated according to other methods available in the literature. Appendix A. Nomenclature

6. Conclusions Two methods based on two different modifications of the capillary bed model are presented for the pressure drop prediction of the flow of purely viscous non-Newtonian fluids through fixed beds of spherical and nonspherical particles along with results of their experimental validation. The results obtained show that both proposed calculation methods give comparable predictions of the pressure drop, and the predicted data are in good agreement with experimental ones for fixed beds of different types of particles in a large range of Reynolds numbers including creeping and inertial flow regimes, and a large range of fluid power-law index (0.269 B nB 1) even when important wall effects exist.

apd aps aw dp Dh Dw fp K lch L Lc Lch m n

dynamic specific surface area, m−1 specific surface of particles (static specific surface area), m−1 specific surface of the apparatus walls, m−1 effective diameter of particles, m apparatus hydraulic diameter, m kinematic variable (Eq. (1)), s−1 friction factor power law model parameter, Pa sn characteristic linear dimension of the capillary cross section, m length of the bed, m the length of the capillary, m characteristic length of the capillary, m number of experimental points power law model parameter

I. Machac' et al. / Chemical Engineering and Processing 37 (1998) 169–176

176

M1 M2 N2 R Repn t u uch V: X

corrective coefficient for the wall effect (Eq. (13)) corrective coefficient of wall effect defined by Eq. (23) corrective coefficient of wall effect defined by Eq. (25) radius of the capillary, m Reynolds number tortuosity superficial velocity, m s−1 characteristic fluid velocity, m s−1 volume flow rate, m3 s−1 fraction of particle surface area defined by Eq. (22)

Greek letters d mean relative deviation di individual relative deviation defined by Eq. (32) DP pressure drop, Pa o bed voidage g; shear rate, s−1 g; pw mean shear rate at the solid surface, s−1 g; w wall shear rate, s−1 8 bed factor r density of the liquid, kg m−3 t shear stress, Pa tpw mean shear stress at the solid surface, Pa tw dynamic variable (Eq. (1)), Pa

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.

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